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2 results

  • FORCING AXIOMS, APPROACHABILITY, AND STATIONARY SET REFLECTION

  • Part of
  • SEAN D. COX
  • Journal:
    The Journal of Symbolic Logic / Volume 86 / Issue 2 / June 2021
    Published online by Cambridge University Press:
    20 October 2021, pp. 499-530
    Print publication:
    June 2021

  • We prove a variety of theorems about stationary set reflection and concepts related to internal approachability. We prove that an implication of Fuchino–Usuba relating stationary reflection to a version of Strong Chang’s Conjecture cannot be reversed; strengthen and simplify some results of Krueger about forcing axioms and approachability; and prove that some other related results of Krueger are sharp. We also adapt some ideas of Woodin to simplify and unify many arguments in the literature involving preservation of forcing axioms.

  • Approachability at the second successor of a singular cardinal

  • Moti Gitik, John Krueger
  • Journal:
    The Journal of Symbolic Logic / Volume 74 / Issue 4 / December 2009
    Published online by Cambridge University Press:
    12 March 2014, pp. 1211-1224
    Print publication:
    December 2009

  • We prove that if μ is a regular cardinal and ℙ is a μ-centered forcing poset, then ℙ forces that (I[μ++[)V generates I[μ++] modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the successor of a singular cardinal.